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For the Gaussian sequence model, we obtain nonasymptotic minimax rates of estimation of the linear, quadratic and the ℓ2-norm functionals on classes of sparse vectors and construct optimal estimators that attain these rates. The main object of interest is the class B0(s) of s-sparse vectors θ = (θ1,..., θd), for which we also provide completely adaptive estimators (independent of s and of the noise variance σ) having logarithmically slower rates than the minimax ones. Furthermore, we obtain the minimax rates on the ℓq-balls Bq(r) = {θ ϵ ℝd : ∥θ∥q ≤ r} where 0 < q ≤ 2, and ${\\Vert \\mathrm{\\theta }\\Vert }_{\\mathrm{q}}={\\left({\\mathrm{\\Sigma }}_{\\mathrm{i}=1}^{\\mathrm{d}}|{\\mathrm{\\theta }}_{\\mathrm{i}}{|}^{\\mathrm{q}}\\right)}^{1/\\mathrm{q}}$. This analysis shows that there are, in general, three zones in the rates of convergence that we call the sparse zone, the dense zone and the degenerate zone, while a fourth zone appears for estimation of the quadratic functional. We show that, as opposed to estimation of θ, the correct logarithmic terms in the optimal rates for the sparse zone scale as log(d/s2) and not as log(d/s). For the class B0(s), the rates of estimation of the linear functional and of the ℓ2-norm have a simple elbow at s = √d (boundary between the sparse and the dense zones) and exhibit similar performances, whereas the estimation of the quadratic functional Q(θ) reveals more complex effects: the minimax risk on B0(s) is infinite and the sparseness assumption needs to be combined with a bound on the ℓ2-norm. Finally, we apply our results on estimation of the ℓ2-norm to the problem of testing against sparse alternatives. In particular, we obtain a nonasymptotic analog of the Ingster–Donoho–Jin theory revealing some effects that were not captured by the previous asymptotic analysis.

In this study, we researched the problem of self-tuning (ST) distributed fusion state estimation for multi-sensor networked stochastic linear discrete-time systems with unknown packet receiving rates, noise variances (NVs), and model parameters (MPs). Packet dropouts may occur when sensor data are sent to a local processor. A Bernoulli distributed stochastic variable is adopted to depict phenomena of packet dropouts. By model transformation, the identification problem of packet receiving rates is transformed into that of unknown MPs for a new augmented system. The recursive extended least squares (RELS) algorithm is used to simultaneously identify packet receiving rates and MPs in the original system. Then, a correlation function method is used to identify unknown NVs. Further, a ST distributed fusion state filter is achieved by applying identified packet receiving rates, NVs, and MPs to the corresponding optimal estimation algorithms. It is strictly proven that ST algorithms converge to optimal algorithms under the condition that the identifiers for parameters are consistent. Two examples verify the effectiveness of the proposed algorithms.

We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector θ ∈ ℝ d belongs to a class of s-sparse vectors with unknown s. We suggest an adaptive estimator achieving a nonasymptotic rate of convergence that differs from the minimax rate at most by a logarithmic factor. We also show that this optimal adaptive rate cannot be improved when s is unknown. Furthermore, we address the issue of simultaneous adaptation to s and to the variance σ² of the noise. We suggest an estimator that achieves the optimal adaptive rate when both s and σ² are unknown.

Cognitive radio, as an important means of implementing spectrum-awareness and dynamic sharing, is of great significance to the future deployment of self-organizing networks. Given its practical application, cognitive radio technology may operate in various adverse self-organizing networks environments. For example, in wireless mobile communication, the multipath propagation with time-varying fading coefficients and unknown noise variance becomes inevitable. Unfortunately, most existing spectrum sensing methods could hardly acquire good performance in this adverse situation. To overcome this difficulty, in this paper we present a novel spectrum sensing algorithm in realistic cognitive radio applications. Firstly, we establish a dynamic state-space model which gives full consideration to the evolution characteristics of primary user’s state and time-variant multipath flat fading channel, while the received signal processed by matched filtering is viewed as the observation. On this basis, spectrum sensing is realized by estimating the primary user’s state, multipath channel impulse response and noise variance jointly and iteratively. This formulated problem is solved based on maximum a posteriori probability criterion and marginal particle filtering technology. Simulations demonstrate that the sensing performance achieved by proposed algorithm is satisfactory and at the same time, the channel response and noise variance are estimated accurately.

Probability hypothesis density (PHD) filter has been demonstrated a promising algorithm for tracking an unknown number of targets in real time. However, this method can only be used in the multi-target tracking systems with known measurement noise variances; otherwise, the tracking performance will decline greatly. To solve this problem, an adaptive PHD filter algorithm is proposed based on the variational Bayesian approximation technique to recursively estimate the joint PHDs of the multi-target states and the time-varying measurement noise variances. First, the variational calculus method is employed to derive the multi-target estimate recursions, and then the Gaussian and the inverse Gamma distributions are introduced to approximate the joint posterior PHD, and achieve a closed-form solution. Simulation results show that the proposed algorithm can effectively estimate the unknown measurement noise variances and has a good performance of multi-target tracking with a strong robustness.

In this paper, the challenging problem of joint channel estimation and data detection for multiple-input multiple-output orthogonal frequency division multiplexing systems operating in time-frequency dispersive channels under unknown background noise is investigated. Based on two different but equivalent signal models, two expectation-maximization algorithm-based iterative schemes for joint data detection and channel and noise variance estimation are proposed. The first scheme jointly detects data and estimates the channel and noise variance, but the computational complexity is high, owing to the simultaneous detection and estimation for all antennas. To reduce the computational complexity, a complexity-reduced scheme that is detecting data and estimating channel for only one antenna during each iteration and holding the unknown quantities of other antennas to their last values is proposed, whose performance only slightly degrades compared to the first scheme. Moreover, both schemes are derived as closed-form expressions, and therefore, our proposed schemes are free of exhaustive search. Simulation results demonstrate quick convergence of the proposed algorithm, and after convergence, the performance of the proposed algorithm is close to that of the optimal channel estimation and data detection case, which assumes full training and perfect channel state information.

Assume that we observe a small set of entries or linear combinations of entries of an unknown matrix A₀ corrupted by noise. We propose a new method for estimating A₀ that does not rely on the knowledge or on an estimation of the standard deviation of the noise ᓂ. Our estimator achieves, up to a logarithmic factor, optimal rates of convergence under Probenius risk and, thus, has the same prediction performance as previously proposed estimators that rely on the knowledge of ó. Some numerical experiments show the benefits of this approach.

This study is concerned with the robust extended Kalman filtering problem for non-linear attitude estimation systems with multiplicative noises and unknown external disturbances. The multiplicative noises are modelled by random variables with bounded variance. The unknown external disturbances are described to lie in bounded set. The objective of the addressed attitude estimation problem is to design a filter such that, in the presence of both the multiplicative noises and unknown external disturbances, an optimised upper bound on the state estimation error variance can be guaranteed. Thus, a robust extended Kalman filter (REKF) is presented for attitude estimation with multiplicative noises and unknown external disturbances. Compared with the traditional extended Kalman filter in attitude estimation, the proposed algorithm takes into consideration the effects of multiplicative noises and unknown external disturbances. Moreover, the stability of the proposed REKF can be proved under certain conditions by utilising the stochastic stability theory. Finally, the simulation results demonstrate the effectiveness of the proposed REKF.

This study deals with the problem of detecting a monochromatic plane wave with unknown amplitude, phase, temporal frequency and direction of arrival in complex white Gaussian noise with unknown variance. Depending on the natural invariance in scale, temporal modulation and spatial modulation of the problem, the uniformly most powerful invariant (UMPI) test is derived by using the statistical invariance principles. However, the UMPI test does not apply unless the signal-to-noise ratio (SNR) is known. However, it provides us with performance bound to evaluate any invariant test's performance when the SNR is unknown. Typically, the generalised likelihood ratio test (GLRT) and locally most powerful invariant (LMPI) test are derived as realisable suboptimal invariant tests with their performance comparison in different SNR through theoretical analysis. Computer simulation examples corroborate the authors analysis and indicate that the GLRT is close to the UMPI bound especially in the low probability-of-false-alarm (PFA) region of the receiver operating characteristic curve while the performance of the LMPI test is close to that of the UMPI test in the low SNR region.

This paper first presents an adaptive expectation-maximization (AEM) control algorithm based on a measurement-data-driven model to reduce the variance of microelectromechanical system (MEMS) accelerometer sensor under multi disturbances. Significantly different characteristics of the disturbances, consisting of drastic-magnitude, short-duration vibration in the external environment, and slowly-varying, long-duration fluctuation inside the sensor are first constructed together with the measurement model of the accelerometer. Next, through establishing a data-driven model based on a historical small measurement sample, the window length of filter of the presented algorithm is adaptively chosen to estimate the sensor state and identify these disturbances simultaneously. Simulation results of the proposed AEM algorithm based on experimental test are compared with the Kalman filter (KF), least mean square (LMS), and regular EM (REM) methods. Variances of the estimated equivalent input under static condition are 0.212 mV, 0.149 mV, 0.015 mV, and 0.004 mV by the KF, LMS, REM, and AEM, respectively. Under dynamic conditions, the corresponding variances are 35.5 mV, 2.07 mV, 2.0 mV, and 1.45 mV, respectively. The variances under static condition based on the proposed method are reduced to 1.9%, 2.8%, and 27.3%, compared with the KF, LMS, and REM methods, respectively. The corresponding variances under dynamic condition are reduced to 4.1%, 70.1%, and 72.5%, respectively. The effectiveness of the proposed method is verified to reduce the variance of the MEMS resonant accelerometer sensor.